I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*, Springer), which I've enjoyed using. I'm confused about a technical hitch though that I'm not sure how to avoid.

The approach taken in this book has two notable simplifying features:

- Studying "matrix Lie groups" (closed subgroups of $\mathrm{GL}_n$) instead of general Lie groups.
- Defining reductive as "is the complexification of the Lie algebra of a compact matrix Lie group" and semisimple as "reductive and center-free." I'll call these compact-reductive and compact-semisimple.

Both of these are nice simplifications that allow one to avoid some technical issues while still getting at the most important material for an introductory course. The latter is very nicely motivated by the question of trying to classify compact Lie groups via Lie algebra theory.

However, one of my students pointed out today a possible problem with combining these two simplifications. It is easy to see that any compact-semisimple Lie algebra is a direct sum of simple Lie algebras, but it is not obvious that these simple summands are themselves compact-semisimple! This means you can't just reduce the classification of compact-semisimple Lie algebras to compact-simple Lie algebras. It seems to me (and this may be my error) that Hall never deals with this issue and tacitly assumes that the simple summands are compact-simple

The obvious theorem to use to get around this issue is the following (Hall Thm 5.11):

Suppose that G is a simply connected matrix Lie group and the Lie algebra $\mathfrak{g}$ of G decomposes as $\mathfrak{g} \cong \mathfrak{h}_1 \oplus \mathfrak{h}_2$. Then there exists closed, simply connected subgroups $H_1$ and $H_2$ of $G$ with Lie algebras $\mathfrak{h}_1$ and $\mathfrak{h}_2$ such that $G = H_1 \times H_2$.

However, there's now a problem, because the compact group $K$ whose complixified Lie algebra of $\mathfrak{g}$ need not have a *matrix* universal cover! (e.g. $\mathrm{SL}_2(\mathbb{R})$). So this argument won't work in Hall's setting.

Is there some simple way to avoid this issue so that one can use both simplifying approaches 1 and 2 simultaneously?

compactmatrix groups do have matrix universal covers, but as far as I can tell that's hard (Peter-Weyl). Another possible approach is to show that the root system breaks up as a direct sum and then use the summand root systems to explicitly realize each of the summands as a compact-semisimple algebra. $\endgroup$ – Noah Snyder Jan 26 '16 at 21:50matrixcompact Lie group? $\endgroup$ – Noah Snyder Jan 26 '16 at 22:54second editionof "Lie groups, Lie algebras, and representations." The whole book has been extensively revised, and a lot of new material added. All the references to theorem and section numbers apply to the second edition only. $\endgroup$ – Brian Hall Mar 13 '16 at 2:47